Properties

Label 2415.1.p.a
Level $2415$
Weight $1$
Character orbit 2415.p
Self dual yes
Analytic conductor $1.205$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -2415
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2415,1,Mod(2414,2415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2415, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2415.2414");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2415.p (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.20524200551\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.5832225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} - q^{3} + ( - \beta + 1) q^{4} - q^{5} + ( - \beta + 1) q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} - q^{3} + ( - \beta + 1) q^{4} - q^{5} + ( - \beta + 1) q^{6} + q^{7} - q^{8} + q^{9} + ( - \beta + 1) q^{10} - \beta q^{11} + (\beta - 1) q^{12} + ( - \beta + 1) q^{13} + (\beta - 1) q^{14} + q^{15} + ( - \beta + 1) q^{17} + (\beta - 1) q^{18} + \beta q^{19} + (\beta - 1) q^{20} - q^{21} - q^{22} + q^{23} + q^{24} + q^{25} + (\beta - 2) q^{26} - q^{27} + ( - \beta + 1) q^{28} + (\beta - 1) q^{30} + q^{32} + \beta q^{33} + (\beta - 2) q^{34} - q^{35} + ( - \beta + 1) q^{36} + (\beta - 1) q^{37} + q^{38} + (\beta - 1) q^{39} + q^{40} + \beta q^{41} + ( - \beta + 1) q^{42} - \beta q^{43} + q^{44} - q^{45} + (\beta - 1) q^{46} + q^{49} + (\beta - 1) q^{50} + (\beta - 1) q^{51} + ( - \beta + 2) q^{52} + ( - \beta + 1) q^{54} + \beta q^{55} - q^{56} - \beta q^{57} + \beta q^{59} + ( - \beta + 1) q^{60} + ( - \beta + 1) q^{61} + q^{63} + (\beta - 1) q^{64} + (\beta - 1) q^{65} + q^{66} + (\beta - 1) q^{67} + ( - \beta + 2) q^{68} - q^{69} + ( - \beta + 1) q^{70} - q^{72} + \beta q^{73} + ( - \beta + 2) q^{74} - q^{75} - q^{76} - \beta q^{77} + ( - \beta + 2) q^{78} + q^{81} + q^{82} + \beta q^{83} + (\beta - 1) q^{84} + (\beta - 1) q^{85} - q^{86} + \beta q^{88} + ( - \beta + 1) q^{90} + ( - \beta + 1) q^{91} + ( - \beta + 1) q^{92} - \beta q^{95} - q^{96} + (\beta - 1) q^{98} - \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} + q^{4} - 2 q^{5} + q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 2 q^{3} + q^{4} - 2 q^{5} + q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + q^{10} - q^{11} - q^{12} + q^{13} - q^{14} + 2 q^{15} + q^{17} - q^{18} + q^{19} - q^{20} - 2 q^{21} - 2 q^{22} + 2 q^{23} + 2 q^{24} + 2 q^{25} - 3 q^{26} - 2 q^{27} + q^{28} - q^{30} + 2 q^{32} + q^{33} - 3 q^{34} - 2 q^{35} + q^{36} - q^{37} + 2 q^{38} - q^{39} + 2 q^{40} + q^{41} + q^{42} - q^{43} + 2 q^{44} - 2 q^{45} - q^{46} + 2 q^{49} - q^{50} - q^{51} + 3 q^{52} + q^{54} + q^{55} - 2 q^{56} - q^{57} + q^{59} + q^{60} + q^{61} + 2 q^{63} - q^{64} - q^{65} + 2 q^{66} - q^{67} + 3 q^{68} - 2 q^{69} + q^{70} - 2 q^{72} + q^{73} + 3 q^{74} - 2 q^{75} - 2 q^{76} - q^{77} + 3 q^{78} + 2 q^{81} + 2 q^{82} + q^{83} - q^{84} - q^{85} - 2 q^{86} + q^{88} + q^{90} + q^{91} + q^{92} - q^{95} - 2 q^{96} - q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2415\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(806\) \(967\) \(1891\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2414.1
−0.618034
1.61803
−1.61803 −1.00000 1.61803 −1.00000 1.61803 1.00000 −1.00000 1.00000 1.61803
2414.2 0.618034 −1.00000 −0.618034 −1.00000 −0.618034 1.00000 −1.00000 1.00000 −0.618034
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
2415.p odd 2 1 CM by \(\Q(\sqrt{-2415}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2415.1.p.a 2
3.b odd 2 1 2415.1.p.f yes 2
5.b even 2 1 2415.1.p.h yes 2
7.b odd 2 1 2415.1.p.d yes 2
15.d odd 2 1 2415.1.p.c yes 2
21.c even 2 1 2415.1.p.g yes 2
23.b odd 2 1 2415.1.p.b yes 2
35.c odd 2 1 2415.1.p.e yes 2
69.c even 2 1 2415.1.p.e yes 2
105.g even 2 1 2415.1.p.b yes 2
115.c odd 2 1 2415.1.p.g yes 2
161.c even 2 1 2415.1.p.c yes 2
345.h even 2 1 2415.1.p.d yes 2
483.c odd 2 1 2415.1.p.h yes 2
805.d even 2 1 2415.1.p.f yes 2
2415.p odd 2 1 CM 2415.1.p.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.1.p.a 2 1.a even 1 1 trivial
2415.1.p.a 2 2415.p odd 2 1 CM
2415.1.p.b yes 2 23.b odd 2 1
2415.1.p.b yes 2 105.g even 2 1
2415.1.p.c yes 2 15.d odd 2 1
2415.1.p.c yes 2 161.c even 2 1
2415.1.p.d yes 2 7.b odd 2 1
2415.1.p.d yes 2 345.h even 2 1
2415.1.p.e yes 2 35.c odd 2 1
2415.1.p.e yes 2 69.c even 2 1
2415.1.p.f yes 2 3.b odd 2 1
2415.1.p.f yes 2 805.d even 2 1
2415.1.p.g yes 2 21.c even 2 1
2415.1.p.g yes 2 115.c odd 2 1
2415.1.p.h yes 2 5.b even 2 1
2415.1.p.h yes 2 483.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2415, [\chi])\):

\( T_{2}^{2} + T_{2} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} + T_{11} - 1 \) Copy content Toggle raw display
\( T_{13}^{2} - T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$13$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$19$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$41$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$43$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$61$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$67$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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